Global Well-Posedness To The Chemotaxis-Fluids Equations In An Unbounded Domain With Boundary

For more than half a century,partial differential equations describing chemotactic phenomena have been increasingly attracted biologists' and mathematicians' attention.In the light of the descriptions of chemotaxis experiments and the background of chemotaxis in real life,this dissertation mainly focuses on the initial-boundary value problems of chemotaxis-fluid coupling models in unbounded or bounded domain with boundaries.The specific research contents are as follows:1.We investigate the initial-boundary value problem of a chemotaxis-Navier-Stokes coupling model with Neumann-Neumann-Dirichlet boundary conditions in an unbounded three-dimensional domain with boundaries.Firstly,by applying the anisotropic Lp inter-polation inequalities and the standard elliptic estimates,we obtain some uniform a priori estimates.Then,by using the continuity arguments,it is proved that the strong solution of the system exists globally and is unique around a constant equilibrium state(n?,0,0),where n? is a nonnegative constant.The key to the proof of this result lies in the careful estimation of the time derivatives and space derivatives,and our results are consistent with the experiment observation and numerical simulation.2.We study the initial-boundary value problem to a chemotaxis-Navier-Stokes cou-pling system with mixed boundary conditions in a 3D unbounded domain with bound-aries.Based on the anisotropic Lp technique,the elliptic estimates and Stokes estimates,we first establish the global existence and uniqueness of strong solution around the equi-librium state(0,csatn,0)with the help of the continuity arguments,where csatn is the sat-uration value of oxygen inside the fluid.Then we use De Giorgi's technique and energy method to show that such a solution will converge to(0,csatn,0)with an explicit conver-gence rate in the chemotaxis-free case.Our assumptions and results are consistent with the experimental descriptions and the numerical analysis,and more closer to our real life.The novelty here consists of deriving some new elliptic estimates and Stokes estimates,and choosing a suitable weight in De Giorgi's technique to deal with the mixed boundary conditions.3.In a 3D bounded domain with smooth boundary,we devote ourselves to the initial-boundary value problem of the Keller-Segel-Stokes coupling model with nonlin-ear diffusion term ?nm(m>0)and rotational flux S(x,n,c).Under the assumption that the Frobenius norm of the tensor-valued chemotactic sensitivity S(x,n,c)satisfies|S(x,n,c)|?CS(1+n)-?,then by seeking some new functionals and using the boot-strap arguments on the regularized system,we establish the existence and boundedness of global weak solutions to the Keller-Segel-Stokes system for arbitrarily large initial data under the assumptions m+2?>2 and m>3/4,which include both the degenerate(m>1)and the singular(m<1)case.